Slides
... % Fill b with square roots of 1 to 1000 using a for loop clear; tic; for i= 1:1000 b(i) = sqrt(i); end t=toc; disp([‘Time taken for loop method is ‘, num2str(t)]); % Fill b with square roots of 1 to 1000 using a vector clear; tic; a = 1:1000; b = sqrt(a); t=toc; disp([‘Time taken for vector method i ...
... % Fill b with square roots of 1 to 1000 using a for loop clear; tic; for i= 1:1000 b(i) = sqrt(i); end t=toc; disp([‘Time taken for loop method is ‘, num2str(t)]); % Fill b with square roots of 1 to 1000 using a vector clear; tic; a = 1:1000; b = sqrt(a); t=toc; disp([‘Time taken for vector method i ...
low-rank matrices with noise and high
... smaller set of macro-variables (combinations of these financial instruments). Similar statements apply to other types of time series data, including neural data [12, 23], subspace tracking models in signal processing and motion models models in computer vision. While the form of system identificatio ...
... smaller set of macro-variables (combinations of these financial instruments). Similar statements apply to other types of time series data, including neural data [12, 23], subspace tracking models in signal processing and motion models models in computer vision. While the form of system identificatio ...
The Smith normal form distribution of a random integer
... If we regard the minors of an n × m matrix as polynomials of the nm matrix entries with integer coefficients, then the SNF of a matrix is uniquely determined by the values of these polynomials. Specifically, let x1 , x2 , . . . , xnm be the nm entries of an n × m matrix, Fj ’s be the minors of an n ...
... If we regard the minors of an n × m matrix as polynomials of the nm matrix entries with integer coefficients, then the SNF of a matrix is uniquely determined by the values of these polynomials. Specifically, let x1 , x2 , . . . , xnm be the nm entries of an n × m matrix, Fj ’s be the minors of an n ...
Sistemi lineari - Università di Trento
... we call the equation system an overdetermined system. Typical applications for such overdetermined systems can be found in data analysis (linear regression) where so-called best-fit curves have to be computed e.g. from observations or experimental data. In this case, the matrix is no longer a N x N ...
... we call the equation system an overdetermined system. Typical applications for such overdetermined systems can be found in data analysis (linear regression) where so-called best-fit curves have to be computed e.g. from observations or experimental data. In this case, the matrix is no longer a N x N ...
Randomized algorithms for matrices and massive datasets
... • Can store large amounts of data, but • Cannot process these data with traditional algorithms. In the Pass-Efficient Model: • Data are assumed to be stored on disk/tape. • Algorithm has access to the data via a pass over the data. • Algorithm is allowed additional RAM space and additional computati ...
... • Can store large amounts of data, but • Cannot process these data with traditional algorithms. In the Pass-Efficient Model: • Data are assumed to be stored on disk/tape. • Algorithm has access to the data via a pass over the data. • Algorithm is allowed additional RAM space and additional computati ...
Fast Monte-Carlo Algorithms for Matrix Multiplication
... • Can store large amounts of data, but • Cannot process these data with traditional algorithms. In the Pass-Efficient Model: • Data are assumed to be stored on disk/tape. • Algorithm has access to the data via a pass over the data. • Algorithm is allowed additional RAM space and additional computati ...
... • Can store large amounts of data, but • Cannot process these data with traditional algorithms. In the Pass-Efficient Model: • Data are assumed to be stored on disk/tape. • Algorithm has access to the data via a pass over the data. • Algorithm is allowed additional RAM space and additional computati ...
SMOOTH ANALYSIS OF THE CONDITION NUMBER AND THE
... entries are iid copies of x, and let M be any deterministic n × n matrix with norm M ≤ nC . Then P(sn (M + Nn ) ≤ n−B ) ≤ n−A . Notice that this theorem requires very little about the variable x. It does not need to be sub-gaussian nor does it have bounded moments. All we ask is that the variance ...
... entries are iid copies of x, and let M be any deterministic n × n matrix with norm M ≤ nC . Then P(sn (M + Nn ) ≤ n−B ) ≤ n−A . Notice that this theorem requires very little about the variable x. It does not need to be sub-gaussian nor does it have bounded moments. All we ask is that the variance ...
Consistency and asymptotic normality
... is available. To understand the principles involved better, we will focus on the case of a scalar regressor xi in this section. In the case of the simple linear model yi = θ0 xi + εi , where xi ∈ R, the closed form solution for the least squares estimator is n X ...
... is available. To understand the principles involved better, we will focus on the case of a scalar regressor xi in this section. In the case of the simple linear model yi = θ0 xi + εi , where xi ∈ R, the closed form solution for the least squares estimator is n X ...
user guide - Ruhr-Universität Bochum
... refinements. When using the Romberg scheme the number given in this field also specifies the number of extrapolation columns. 4. Zeros of functions This method demonstrates several algorithms for computing real zeros of functions of one real variable. The choice Function provides the following sampl ...
... refinements. When using the Romberg scheme the number given in this field also specifies the number of extrapolation columns. 4. Zeros of functions This method demonstrates several algorithms for computing real zeros of functions of one real variable. The choice Function provides the following sampl ...
On Multiplicative Matrix Channels over Finite Chain
... A ring R is called a chain ring if, for any two ideals I, J of R, either I ⊆ J or J ⊆ I. It is known that a finite ring R is a chain ring if and only if R is both principal (i.e., all of its ideals are generated by a single element) and local (i.e., the ring has a unique maximal ideal). Let π ∈ R be ...
... A ring R is called a chain ring if, for any two ideals I, J of R, either I ⊆ J or J ⊆ I. It is known that a finite ring R is a chain ring if and only if R is both principal (i.e., all of its ideals are generated by a single element) and local (i.e., the ring has a unique maximal ideal). Let π ∈ R be ...
Using Mixture Models for Collaborative Filtering.
... to ours, both because of these differences in the underlying generative model, as well as differences in the objective function and the way in which data is gathered from users. We discuss this comparison further below, focusing on the relationship between the spectral methods employed by [2, 4] and ...
... to ours, both because of these differences in the underlying generative model, as well as differences in the objective function and the way in which data is gathered from users. We discuss this comparison further below, focusing on the relationship between the spectral methods employed by [2, 4] and ...
Matrices and RRE Form Notation. R is the real numbers, C is the
... To prove this theorem, it suffices (by induction) to verify it when B is obtained from A by a single elementary row operation. A linear system of equations has a unique solution, infinitely many solutions, or no solutions (the system is inconsistent). A homogeneous linear system of equations always ...
... To prove this theorem, it suffices (by induction) to verify it when B is obtained from A by a single elementary row operation. A linear system of equations has a unique solution, infinitely many solutions, or no solutions (the system is inconsistent). A homogeneous linear system of equations always ...
6.4 Krylov Subspaces and Conjugate Gradients
... always, those are the squares of the singular values αmax and αmin of Vc . The condition number of the power basis 1, x, x2 , x3 is the ratio αmax /αmin � 125. If you want a more impressive number (a numerical disaster), go up to x9 . The condition number of the 10 by 10 Hilbert matrix is �max /�min ...
... always, those are the squares of the singular values αmax and αmin of Vc . The condition number of the power basis 1, x, x2 , x3 is the ratio αmax /αmin � 125. If you want a more impressive number (a numerical disaster), go up to x9 . The condition number of the 10 by 10 Hilbert matrix is �max /�min ...
0 jnvLudhiana Page 1
... Degree of differential equation The Degree of the differential equation is the index of the highest order derivatives which appears in the differential equation after making it free from negative and fractional ...
... Degree of differential equation The Degree of the differential equation is the index of the highest order derivatives which appears in the differential equation after making it free from negative and fractional ...
Cubic Spline Interpolation of Periodic Functions
... there is no need to derive everything from scratch. In fact, if you answered (A), you can avoid doing any calculations, just look at the equations for the spline coefficient at the internal points for the case of the free or clamped splines. (C) The n×n matrix A is “almost” tridiagonal – its only en ...
... there is no need to derive everything from scratch. In fact, if you answered (A), you can avoid doing any calculations, just look at the equations for the spline coefficient at the internal points for the case of the free or clamped splines. (C) The n×n matrix A is “almost” tridiagonal – its only en ...
Sparse Matrices and Their Data Structures (PSC §4.2)
... nonzero in the same row and column. Offers maximum flexibility: row-wise and column-wise access are easy and elements can be inserted and deleted in O(1) operations. ...
... nonzero in the same row and column. Offers maximum flexibility: row-wise and column-wise access are easy and elements can be inserted and deleted in O(1) operations. ...
Slide 1
... Each nonzero value of x3 determines a nontrivial solution of (1). Hence, v1, v2, v3 are linearly dependent. Slide 1.7- 5 ...
... Each nonzero value of x3 determines a nontrivial solution of (1). Hence, v1, v2, v3 are linearly dependent. Slide 1.7- 5 ...
Lecture 2. Solving Linear Systems
... make sure verify your answer by direct multiplication of LU: 5. For each of the following statements, determine whether it is true or false. If your answer is true, state your rationale. If false, provide an counterexample (the example contradicting the statement). (a) A matrix may be row reduced to ...
... make sure verify your answer by direct multiplication of LU: 5. For each of the following statements, determine whether it is true or false. If your answer is true, state your rationale. If false, provide an counterexample (the example contradicting the statement). (a) A matrix may be row reduced to ...
Vector Norms
... In computing the solution to any mathematical problem, there are many sources of error that can impair the accuracy of the computed solution. The study of these sources of error is called error analysis, which will be discussed later in this lecture. First, we will focus on one type of error that oc ...
... In computing the solution to any mathematical problem, there are many sources of error that can impair the accuracy of the computed solution. The study of these sources of error is called error analysis, which will be discussed later in this lecture. First, we will focus on one type of error that oc ...
4 Singular Value Decomposition (SVD)
... provided that for each i > 1, σi (A) < σ1 (A). This suggests a way of finding σ1 and u1 , by successively powering B. But there are two issues. First, if there is a significant gap between the first and second singular values of a matrix, then the above argument applies and the power method will qui ...
... provided that for each i > 1, σi (A) < σ1 (A). This suggests a way of finding σ1 and u1 , by successively powering B. But there are two issues. First, if there is a significant gap between the first and second singular values of a matrix, then the above argument applies and the power method will qui ...
Hotelling`s One
... This option specifies one or more values for the probability of a type-I error. A type-I error occurs when a true null hypothesis is rejected. In this procedure, a type-I error occurs when you reject the null hypothesis of equal means when in fact the means are equal. Values must be between zero and ...
... This option specifies one or more values for the probability of a type-I error. A type-I error occurs when a true null hypothesis is rejected. In this procedure, a type-I error occurs when you reject the null hypothesis of equal means when in fact the means are equal. Values must be between zero and ...
rotations: An R Package for SO(3) Data
... preserve the geometry of SO(3). These can be computed using the mean and median functions with the argument type = "geometric". Table 2 summarizes the four estimators including their formal definition and how they can be computed. The estimators in Table 2 find estimates based on minimization of L1 ...
... preserve the geometry of SO(3). These can be computed using the mean and median functions with the argument type = "geometric". Table 2 summarizes the four estimators including their formal definition and how they can be computed. The estimators in Table 2 find estimates based on minimization of L1 ...
Subspace Embeddings for the Polynomial Kernel
... embedding, that is, simultaneously for all v ∈ V , kφ(v) · Sk2 = (1 ± )kφ(v)k2 . T ENSOR S KETCH can be seen as a very restricted form of C OUNT S KETCH, where the additional restrictions enable its fast running time on inputs which are tensor products. In particular, the hash functions in T EN SOR ...
... embedding, that is, simultaneously for all v ∈ V , kφ(v) · Sk2 = (1 ± )kφ(v)k2 . T ENSOR S KETCH can be seen as a very restricted form of C OUNT S KETCH, where the additional restrictions enable its fast running time on inputs which are tensor products. In particular, the hash functions in T EN SOR ...
Ordinary least squares
In statistics, ordinary least squares (OLS) or linear least squares is a method for estimating the unknown parameters in a linear regression model, with the goal of minimizing the differences between the observed responses in some arbitrary dataset and the responses predicted by the linear approximation of the data (visually this is seen as the sum of the vertical distances between each data point in the set and the corresponding point on the regression line - the smaller the differences, the better the model fits the data). The resulting estimator can be expressed by a simple formula, especially in the case of a single regressor on the right-hand side.The OLS estimator is consistent when the regressors are exogenous and there is no perfect multicollinearity, and optimal in the class of linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances. Under the additional assumption that the errors be normally distributed, OLS is the maximum likelihood estimator. OLS is used in economics (econometrics), political science and electrical engineering (control theory and signal processing), among many areas of application. The Multi-fractional order estimator is an expanded version of OLS.